## (a) Show that for a projectile the angle between the velocity

[question] Question. (a) Show that for a projectile the angle between the velocity and the x-axis as a function of time is given by $\theta(t)=\tan ^{-1}\left(\frac{v_{0 y}-\mathrm{g} t}{v_{0 x}}\right)$ (b) Show that the projection angle $\theta_{0}$ for a projectile launched from the origin is given by $\theta_{0}=\tan ^{-1}\left(\frac{4 h_{m}}{R}\right)$ Where the symbols have their usual meaning. [/question] [solution] solution: (a) Let $v_{0 x}$ and $v_{0 y}$ respectively be the initial com...

## Can you associate vectors with (a) the length of a wire bent into a loop,

[question] Question. Can you associate vectors with (a) the length of a wire bent into a loop, (b) a plane area, (c) a sphere? Explain. [/question] [solution] solution: Answer: No; Yes; No (a) One cannot associate a vector with the length of a wire bent into a loop. (b) One can associate an area vector with a plane area. The direction of this vector is normal, inward or outward to the plane area. (c) One cannot associate a vector with the volume of a sphere. However, an area vector can be associ...

## A vector has magnitude and direction.

[question] Question. A vector has magnitude and direction. Does it have a location in space? Can it vary with time? Will two equal vectors a and b at different locations in space necessarily have identical physical effects? Give examples in support of your answer. [/question] [solution] solution: Answer: No; Yes; No Generally speaking, a vector has no definite locations in space. This is because a vector remains invariant when displaced in such a way that its magnitude and direction remain the s...

## A stone tied to the end of a string 80 cm long is whirled in a

[question] Question. A stone tied to the end of a string 80 cm long is whirled in a horizontal circle with a constant speed. If the stone makes 14 revolutions in 25 s, what is the magnitude and direction of acceleration of the stone? [/question] [solution] solution: Length of the string, $I=80 \mathrm{~cm}=0.8 \mathrm{~m}$ Number of revolutions $=14$ Frequency, $v=\frac{\text { Number of revolutions }}{\text { Time taken }}=\frac{14}{25} \mathrm{~Hz}$ Angular frequency, $\omega=2 \pi \mathrm{V}$...

## Rain is falling vertically with a speed of

[question] Question. Rain is falling vertically with a speed of $30 \mathrm{~m} \mathrm{~s}^{-1}$. A woman rides a bicycle with a speed of $10 \mathrm{~m} \mathrm{~s}^{-1}$ in the north to south direction. What is the direction in which she should hold her umbrella? [/question] [solution] solution: The described situation is shown in the given figure. Here, $v_{c}=$ Velocity of the cyclist $v_{\mathrm{r}}=$ Velocity of falling rain In order to protect herself from the rain, the woman must hold h...

## Given $\mathbf{a}+\mathbf{b}+\mathbf{c}+\mathbf{d}=0$, which of the following statements are correct:

[question] Question. Given $\mathbf{a}+\mathbf{b}+\mathbf{c}+\mathbf{d}=0$, which of the following statements are correct: (a) $\mathbf{a}, \mathbf{b}, \mathbf{c}$, and $\mathrm{d}$ must each be a null vector, (b) The magnitude of $(\mathbf{a}+\mathbf{c})$ equals the magnitude of $(\mathbf{b}+\mathbf{d})$, (c) The magnitude of a can never be greater than the sum of the magnitudes of $\mathbf{b}, \mathbf{c}$, and $\mathbf{d}$, (d) $\mathbf{b}+\mathbf{c}$ must lie in the plane of $\mathbf{a}$ and ...

## Establish the following vector inequalities geometrically or otherwise:

[question] Question. Establish the following vector inequalities geometrically or otherwise: (a) $|\mathbf{a}+\mathbf{b}| \leq|\mathbf{a}|+|\mathbf{b}|$ (b) $|\mathbf{a}+\mathbf{b}| \geq\|\mathbf{a}|-| \mathbf{b}\|$ (c) $|\mathbf{a}-\mathbf{b}| \leq|\mathbf{a}|+|\mathbf{b}|$ (d) $|\mathbf{a}-\mathbf{b}| \geq|| \mathbf{a}|-| \mathbf{b}||$ When does the equality sign above apply? [/question] [solution] solution: (a) Let two vectors $\vec{a}$ and $\vec{b}$ be represented by the adjacent sides of a ...

## Pick out the only vector quantity in the following list: Temperature, pressure,

[question] Question. Pick out the only vector quantity in the following list: Temperature, pressure, impulse, time, power, total path length, energy, gravitational potential, coefficient of friction, charge. [/question] [solution] solution: Impulse Impulse is given by the product of force and time. Since force is a vector quantity, its product with time (a scalar quantity) gives a vector quantity. [/solution]...

## Pick out the two scalar quantities in the following list:

[question] Question. Pick out the two scalar quantities in the following list: force, angular momentum, work, current, linear momentum, electric field, average velocity, magnetic moment, relative velocity. [/question] [solution] solution: Work and current are scalar quantities. Work done is given by the dot product of force and displacement. Since the dot product of two quantities is always a scalar, work is a scalar physical quantity. Current is described only by its magnitude. Its direction is...

## Pick out the two scalar quantities in the following list:

[question] Question. Pick out the two scalar quantities in the following list: force, angular momentum, work, current, linear momentum, electric field, average velocity, magnetic moment, relative velocity. [/question] [solution] solution: calar: Volume, mass, speed, density, number of moles, angular frequency Vector: Acceleration, velocity, displacement, angular velocity A scalar quantity is specified by its magnitude only. It does not have any direction associated with it. Volume, mass, speed, ...

## State, for each of the following physical quantities,

[question] Question. State, for each of the following physical quantities, if it is a scalar or a vector: volume, mass, speed, acceleration, density, number of moles, velocity, angular frequency, displacement, angular velocity. [/question] [solution] solution: Scalar: Volume, mass, speed, density, number of moles, angular frequency Vector: Acceleration, velocity, displacement, angular velocity A scalar quantity is specified by its magnitude only. It does not have any direction associated with it...